1/44

Title: Immunological Self-Tolerance: Lessons from Mathematical Modeling

Authors: Jorge Carneiro1, Tiago Paixão1, Dejan Milutinovic2, João Sousa1, Kalet Leon1,3,

Rui Gardner1, and Jose Faro1

Affiliation:

1. Instituto Gulbenkian de Ciência, Oeiras

2. Instituto Superior Técnico, Lisboa

3. Centro de Inmunologia Molecular, Habana

Correspondence to:

Jorge Carneiro

Instituto Gulbenkian de Ciencia

Apartado 14

2781-901 Oeiras

Portugal

Phone: (351) 214 407 920

Fax: (351) 214 407 973

Email: jcarneir@igc.gulbenkian.pt

2/44

Abstract

One of the fundamental properties of the immune system is its capacity to avoid autoimmune

diseases. The mechanism underlying this process, known as self-tolerance, is hitherto

unresolved but seems to involve the control of clonal expansion of autoreactive lymphocytes.

This article reviews mathematical modeling of self-tolerance, addressing two specific

hypotheses. The first hypothesis posits that self-tolerance is mediated by tuning of activation

thresholds, which makes autoreactive T lymphocytes reversibly “anergic” and unable to

proliferate. The second hypothesis posits that the proliferation of autoreactive T lymphocytes

is instead controlled by specific regulatory T lymphocytes. Models representing the

population dynamics of autoreactive T lymphocytes according to these two hypotheses were

derived. For each model we identified how cell density affects tolerance, and predicted the

corresponding phase spaces and bifurcations. We show that the simple induction of

proliferative anergy, as modeled here, has a density dependence that is only partially

compatible with adoptive transfers of tolerance, and that the models of tolerance mediated by

specific regulatory T cells are closer to the observations.

3/44

1. Introduction

Mathematical modeling of the immune system often concentrates on the immune responses to

pathogens. This article deals with another fundamental process in the immune system: the

maintenance of self-tolerance, i.e., the prevention of harmful immune responses against body

components. The biological significance of this process becomes very patent upon its failure

during pathological conditions known as autoimmune diseases.

The risk of autoimmunity cannot be dissociated from the capacity of the immune

system to cope with diverse and fast evolving pathogens (Langman and Cohn, 1987). The

latter is achieved by setting up a vast and diverse repertoire of antigen receptors expressed by

lymphocytes, which as a whole is capable of recognizing any possible antigen. Most

lymphocytes have a unique antigen receptor (immunoglobulin in B-cells and TCR in T cells)

that is encoded by a gene that results from somatic mutation and random assortment of gene

segments in lymphocyte precursors. The randomness in the generation of antigen receptors

makes it unavoidable that lymphocytes with receptors recognizing body antigens are also

made. These autoreactive lymphocytes can potentially cause autoimmune diseases if their

activation and clonal expansion is not prevented. The question is how is this avoided in

healthy individuals?

According to Burnet’s original clonal selection theory (Burnet, 1957) expansion of

autoreactive lymphocytes and autoimmunity would be avoided by deleting the autoreactive

lymphocytes from the repertoire once and for all during embryonic development. The fact that

the generation of lymphocytes is a life long process in mammals invalidated this possibility.

Following an early suggestion by Lederberg (Lederberg, 1959) deletion of potentially self-

destructive lymphocytes was reformulated as an aspect of lymphopoiesis. Accordingly,

lymphocytes that express an autoreactive receptor are deleted at an immature stage of their

4/44

development (Kisielow et al., 1988; Acha-Orbea and MacDonald, 1995), before they can

undergo clonal expansion and trigger destructive immune responses.

But deletion alone cannot explain self-tolerance. The major shortcoming of deletion

models of self-tolerance is the well-documented presence of mature autoreactive B and T

lymphocytes in normal healthy animals (Pereira et al., 1986). Many different experiments

have demonstrated that these autoreactive T cells can undergo clonal expansion and cause

disease. In this paper we will focus on a particular type of experiments first reported by

Sakaguchi and coworkers (Sakaguchi et al., 1995; Sakaguchi et al., 2001), which allows

assessing self-tolerance from the perspective of the population dynamics of circulating

lymphocytes. CD4+ T cells were isolated from healthy animals and subsets of this population

were transferred into syngeneic recipient animals, which were devoid of T cells. Transfer of

CD4+CD25- T cells resulted in the expansion of these cells in the recipients and caused an

autoimmune syndrome characterized by multiple organ-specific autoimmune diseases

(illustrated in fig. 1). These results indicate that in the healthy individuals there are significant

numbers of autoreactive cells that could potentially proliferate and mount deleterious immune

responses to self.

How are those autoreactive T cells, circulating in healthy individuals, prevented from

mounting harmful immune responses against body tissues? There are several hypotheses in

the literature (see the special issue of Seminars Immunology (vol 12 issue 3) for a rather

comprehensive overview). One hypothesis posits that autoreactive T cells are prevented from

proliferating and mounting immune responses because specific regulatory T cells control

them. In the above-mentioned Sakaguchi et al. experiment (fig. 1), those animals devoid of T

cells receiving the same number of CD4+CD25+ T cells or receiving equal numbers of

CD4+CD25- and CD4+CD25+ T cells did not develop autoimmune diseases. Prevention of

autoimmunity in the recipients by transfer of CD25+ T cells suggests the existence of

5/44

regulatory T cells within the CD25+ subset, which exert a direct suppressive interaction on

CD25- T cells. Although this interpretation has been favored by immunologists, recent

evidence suggests that competition and density-dependent inhibition of cell expansion in

recipients may be sufficient to explain the inhibitory effects of CD4+CD25+ cells on

CD4+CD25- cells (Barthlott et al., 2003), and thus postulating direct suppressive effects could

be superfluous. Another hypothesis for the prevention of harmful immune responses by

autoreactive T cells is that these cells become unresponsive to self-antigens by modification of

their cell-signaling machinery; immunologists use the word “anergy” to refer to this

unresponsiveness of cells, namely when this is reflected in diminished proliferative responses

(Schwartz, 1990). Among the possible explanations for self-specific anergy induction, perhaps

the simplest is the hypothesis that lymphocytes tune up their activation thresholds in response

to recurrent stimuli (Grossman and Paul, 1992; Grossman and Singer, 1996; Grossman and

Paul, 2000; Grossman and Paul, 2001). According to this tunable activation threshold (TAT)

hypothesis, autoreactive lymphocytes that are frequently stimulated by particular self-antigens

adapt to the recent time-average of such stimulation so that they fail to be activated by these

antigens.

We have previously addressed these two hypotheses by modeling the population

biology of autoreactive T lymphocytes. In this article we review our published results and

present new ones obtained with two simplified models of autoreactive T cell dynamics.

Basically, we ask here whether and to what extent the underlying mechanisms of tolerance

induction and maintenance within each model are compatible with the basic aspects of the

Sakaguchi phenomenon (fig. 1). This phenomenon is particularly suitable for modeling since

in these experiments the control of autoimmunity can be understood as the control of

proliferation, in the absence of thymic influx that would otherwise complicate the

mathematics. In the next section we propose and analyse a hypothesis according to which

6/44

recurrent stimulation by self-antigens and tuning of activation thresholds regulate the

proliferative responses of autoreactive T lymphocytes. We show that this induction of

proliferative anergy in autoreactive T cells has a density dependence that is only partially

compatible with the Sakaguchi phenomenon. In contrast, as shown in section 3, our model of

regulation of activation-dependent proliferation of autoreactive T cells by specific regulatory

T cells results in realistic density dependence.

7/44

2. Modeling tolerance by tuning of activation thresholds of individual T lymphocytes

The tunable activation threshold (TAT) hypothesis by Grossman et al. (Grossman and Paul,

1992; Grossman and Singer, 1996; Grossman and Paul, 2000; Grossman and Paul, 2001)

proposes that every interaction between the TCR and its ligand on APCs results in an

intracellular competition between "excitation" and "de-excitation" signaling pathways that

causes the T cell to adapt to the stimulation by increasing or decreasing its threshold for

activation. Recently, this hypothesis has been discussed by other authors in the context of

peripheral tolerance, and T cell proliferation and homeostasis (Nicholson et al., 2000; Smith et

al., 2001; Tanchot et al., 2001; Wong et al., 2001; Singh and Schwartz, 2003). Mathematical

models of adaptation of neural synapses would be a straightforward inspiration for modeling

the TAT mechanism. However, the set up of the immune system poses specific problems. In

neural tissue intermittent signaling does not depend on de novo formation of synapses, which

are sessile. In contrast, the immunological synapses are intermittent and their formation

depends on the relative densities of T cells and APCs bearing agonist antigens. Synapse

formation and signaling is therefore coupled to population dynamics, which, as we will see

below, poses novel mathematical modeling problems. Grossman and Paul (1992, 2001)

extensively discuss the coupling of tuning and population dynamics but an explicit

mathematical model was not put forward. The model presented here features tuning of the

signaling machinery of individual T cells in close agreement with what was proposed by

Grossman and Paul (2001). However, the way this signaling is coupled to the population

dynamics departs from the proposal of these authors. Here, adaptation is restricted to the

activation-dependent proliferative response, whereas the general tuning hypothesis by

Grossman and colleagues is more comprehensive, contemplating features such as the role of

suppression in inducing tuning and vice-versa, and tuning of APCs. In fact, Grossman and

8/44

colleagues did not propose that TAT per se would be responsible for regulating the expansion

and maintenance of T cell populations as we are studying here.

A MINIMAL MODEL

Activation, proliferation and survival of T lymphocytes require recurrent interactions of their

TCRs with their ligands, the MHC-peptide complexes, at the membrane of antigen presenting

cells (APCs) (Witherden et al., 2000; Polic et al., 2001). This APC-dependent population

dynamics is captured in the reactional diagram proposed by De Boer and Perelson (De Boer

and Perelson, 1994) (fig. 2A). Assuming that the densities of the conjugated T cells are in

quasi-steady state (De Boer and Perelson, 1994; Sousa, 2003), this diagram is translated into

the following differential equation (see Appendix A for derivation):

†

dTdt

= daC -d(T - C) (1)

where T is the T cell density, d is the conjugate dissociation rate, d is the per cell death rate,

and C is the quasi-steady state conjugate density:

†

C @c T + A( ) + d - -4 ⋅ A ⋅ T ⋅ c 2 + c T + A( ) + d( )2

2c(2)

According to the TAT hypothesis, continuous signaling by the TCR would lead to

dynamic adaptation of the signal transduction machinery. To incorporate this in the model

above we must define the probability a of productive conjugation as a function of the

signaling machinery status of the conjugated lymphocyte. Following closely the conceptual

signaling model by Grossman and Paul (1992, 2001), Sousa et al. (Sousa, 2003) assumed that

T cell activation is controlled by a “futile cycle” downstream of TCR signaling, involving a

kinase and a phosphatase that operate on an adapter molecule (Clements et al., 1999; Germain

and Stefanova, 1999; Stefanova et al., 2003) (fig. 2). Also as suggested by Grossmann and

9/44

Paul (1992, 2001), it was assumed that the phosphorylation state of the adapter is

hypersensitive to the relative activities of the two enzymes and behaves as a molecular switch

(Koshland et al., 1982). All the adapter molecules are phosphorylated if the kinase activity is

higher than that of the phosphatase; otherwise, the adapter is fully dephosphorylated.1 During

lymphocyte conjugation with an APC, TCR stimuli result in a faster increase in kinase and

phosphatase activities. The lymphocyte will be activated and enter cell cycle if after

conjugating with the APC the kinase activity supersedes that of the phosphatase, and it will

remain quiescent otherwise. Note that by linking activation to the function of proliferation

alone we depart from Grossman and Paul (2001), who discuss other functions relevant to

population dynamics as well. The dynamics of this signaling machinery was represented by

two differential equations:

†

dKdt

= rK K0(1+ s) - K( ) (3)

†

dPdt

= rP P0(1+ s) - P( ) (4)

where K is the kinase activity, P is the phosphatase activity, rK is the turnover rate of the

kinase, rP is the turnover rate of the phosphatase, K0 is the basal steady state kinase activity,

and P0 is the basal steady state phosphatase activity. Parameter s is the magnitude of the

stimulus to the kinase and phosphatase production rates, which takes the value 0 if the cell is

free and s if conjugated. This signaling cascade shows adaptive properties (Grossman and

Paul, 2001) provided that the turnover rate of the kinase is higher than that of the phosphatase

(rk>rP), and that, for any stimuli, the steady state activity of the phosphatase is higher than

that of the kinase (P0>K0). Under these conditions, the adapter can be transiently switched on,

but it will be switched off eventually if the stimulus persists.

This simple mathematical TAT model was developed and analysed in Sousa et al. 1 A very similar adaptation model was developed and analysed by Levchenko and Iglesias(2002) in the context of gradient sensing.

10/44

(Sousa, 2003), based mainly on Monte-Carlo stochastic simulations of individual cells. In this

article we present a further simplification, which is amenable to analytic treatment and retains

the main properties of the original model. This simplification involves two additional

approximations. First, we assume that turnover of the kinase activity is very fast as compared

to the conjugate dissociation rate (rK>>d), and as compared to the turnover rate of the

phosphatase activity (rK>>rP). Under these conditions, the kinase activity is in quasi-steady

state, and can be approximated by either K = K0(1+s) or K = K0, respectively, when the T

lymphocyte is conjugated to an APC, resulting in a stimulus s, or when the lymphocyte is

free, resulting in no stimulus. The second approximation consists of assuming that for any

given density of T cells and APCs, the fast conjugation and deconjugation processes are

practically in equilibrium. This implies that the probability density functions (PDFs) of the

phosphatase activity in conjugated and in free T cell populations are stationary. These

approximations were mainly motivated by the simplicity they confer to the mathematics. The

first assumption can be biologically sustained since very early events seem to define whether

or not a cell will be activated. As for the second assumption, it is not guaranteed that it holds

in a growing population since it is unlikely that the distribution of the phosphatase activity

will become stationary before the size of the population changes. However, it will obviously

hold at the equilibrium, and therefore it is safe to draw qualitative conclusions from this

simplified model in terms of number and stability of its steady states. Our confidence is

further supported by the fact that the same qualitative results were obtained with more

realistic Monte-Carlo simulations of individual cells where these simplifying assumptions

were not introduced (Sousa, 2003).

Milutinovic et al. (2004) used stochastic hybrid-automaton theory to describe the

PDFs of cell-associated molecules in cells that cycle between APC-conjugation and APC-free

states. The dynamics of phosphatase PDF in conjugated and free T cells, respectively rC and

11/44

rF, are described by the following set of first order partial differential equations:

†

∂rC

∂t+

∂∂P

(PC rC ) = -drC + cE rF (5)

†

∂rF

∂t+

∂∂P

(PF rF ) = drC - cE rF (6)

where PC and PF are the functions governing the dynamics of the phosphatase in the

conjugated and free regimes (i.e. the right hand side of eqn. 4 with s>0 and s=0, respectively

rP(P0(1+s)-P) and rP(P0-P)), and cE and d are the transition rates from the free to the

conjugated state (cE=c(A-C)) and from the conjugated to the free state, respectively.

Assuming that the conjugation and deconjugation processes are in quasi-steady state,

we expect the time derivatives to vanish. Under these conditions, we obtain the following

equation:

†

∂∂P

(PC rC + PF rF ) = 0 (7)

which, as demonstrated by Milutinovic et al. (2004), can be used to reduce the system to the

following differential equation:

†

∂∂P

(PC rC ) = - d + cEPC

PF

Ê

Ë Á

ˆ

¯ ˜ rC (8)

The solution of this equation is:

†

rC = N P0(1+ s ) - PdrP

-1 P0 - PcErP , P0 £ P £ P0(1+ s )

0 , else

Ï Ì Ô

Ó Ô (9)

where N is a normalisation constant (a detailed derivation of equation 9 is provided in

Appendix B).

The fraction a of T cells that is activated and divides when conjugation ceases (i.e. the

fraction of T cells that at the instant of releasing from the APC have K>P) is then:

†

a = rC dPP0

K0 (1+s )Ú (10)

12/44

Substituting this definition in eqn. 1 we fully define the population dynamics of the T

cells with TAT.

RESULTS

The (K,P)-signaling machinery shows an activation threshold that is tunable. It is easy

to note that if the value of the phosphatase activity at the beginning of conjugation is

†

P ≥ K0(1+ s ) then this is sufficient (although not necessary) to prevent activation of the T

cell. The threshold is modulated by the history of stimuli to the T cell, which determines the

value of the phosphatase activity P at any given time. Therefore, from the point of view of the

population biology of T cells, in this model, as in the conceptual model of Grossman and Paul

(2001), the activation threshold is dependent on the frequency of interactions of T cells with

the APCs, i.e. the frequency of the stimuli to the individual T cells (Sousa, 2003) (fig. 3A). If

T cells were always free, the probability density function of the phosphatase activity would

correspond to a Dirac Delta at P0; whereas if all the lymphocytes were permanently

conjugated to APCs delivering the same stimulus s then the PDF would be a Dirac Delta at

P0(1+ s ) (fig. 3B). Since T cells are cycling between conjugation and free periods, the

stationary PDF of P in the population takes values in the interval [P0 , P0(1+ s)] (Appendix

B).

The frequency of APC interactions per T cell decreases as T cell density increases due

to competition (eqn 2). This implies that, as T cell density increases, the median of the PDF of

phosphatase activity in conjugated T cells (rc(P)) becomes closer to the value P 0 ;

reciprocally, as T cell density decreases, the median of the PDF approaches P0(1 + s)

(fig.3B). This means that the fraction of cells a undergoing productive conjugation to

activation and cell cycle increases with T cell density. This defines a positive feedback loop

13/44

such that increases (decreases) in T cell density result in higher (lower) average values of a,

which lead to further increases (decreases) in T cell density. This positive feedback loop

resulting from the present implementation of tunable activation thresholds is the opposite of a

density-dependent feedback population control. In our model this loop interacts with the

negative feedback loop defined by the effect of competition on the density of conjugates. For

this reason, the model has two possible stable steady states: one in which lymphocyte

population is extinct and one in which it is limited by APC availability, and predominantly

made of non-anergic lymphocytes (fig.3C). The bifurcation diagram (fig.3D) of the steady

state population size as a function of the ratio P0/K0, which is a measure of the adaptability of

the signaling cascade, indicates that the main contribution of tunable thresholds at

intermediate P0/K0 values is to change the size of the basins of attraction of the extinction and

APC-limited states by shifting the position of the saddle point (actually in a model without

adaptation there is no saddle point and the extinction state is unstable (De Boer and Perelson,

1994; Sousa, 2003)); as this control parameter increases the size of the population in the

saddle point, and decreases that of the APC-limited state, there is a fold-bifurcation at a

critical value of this parameter in which both points merge into a single one. Beyond this

critical value such points disappear. In this mathematical TAT model the only way the

population of autoreactive T cells can persist is by competition for limited numbers of APCs;

if TAT effects predominate the population will become extinct.

SPECIFIC DISCUSSION

Our model in which the TAT-signaling machinery is coupled to the growth dynamics of T cell

populations offers a mechanism of self-tolerance by prevention of autoreactive T cell

expansion, and their eventual deletion from the circulating pool. Therefore, according to this

14/44

model, the persistence of circulating anergic T cells requires their continuous influx from the

thymus, as demonstrated by Sousa et al. (Sousa, 2003). This is not unreasonable because the

thymus continuously produces T cells although at a rate that decreases with age. However,

when confronted with the Sakaguchi phenomenon, which involves adoptive transfers of

peripheral T cells in the absence of thymic influx, the present mathematical model shows

some important limitations.

Within the framework of our model, an adoptive transfer procedure in which

lymphocytes are isolated ex vivo can be interpreted as an extra time-period during which the

transferred lymphocytes remain free from the APCs. This manipulation would increase the

responsiveness of lymphocytes as compared to the steady state in vivo, and this would be

compatible with the experimental observations that CD4+CD25- lymphocytes from healthy

subjects can induce autoimmunity in empty recipients. The fact that CD4+CD25+ T cells do

not induce autoimmune disease in the recipient animals can also be interpreted by assuming

that CD25+ cells have higher thresholds of activation (higher values of phosphatase activity

P) than CD25- cells. This is not unreasonable because it is well documented that, following

anergy induction in vitro, lymphocytes upregulate the CD25 molecule (Kuniyasu et al., 2000).

However, the observation that cotransfers of CD4+CD25- and CD4+CD25+ cells result in

tolerance cannot be interpreted within the framework of our simple mathematical TAT model

alone. As we have demonstrated, in our model "more cells should lead to more responsiveness

or less anergy", and therefore adding CD25+ T cells to CD25- cells should have increased the

proliferative responsiveness of the CD25- cells rather than suppress it and change the total

population steady state as observed (Annacker et al., 2001). The result of the co-transfers thus

point to the existence of some kind of interaction between cell populations that was not taken

into account in our simple mathematical model. A suppressive interaction among T cells will

be explicitly discussed and modeled in the next section.

15/44

As mentioned above, the model we presented here was designed to include the

properties of tuning of activation thresholds posited by Grossman and Paul (2001) but, by

coupling tuning to the proliferative response alone, it departs significantly from the more

comprehensive conceptual hypothesis that these authors have put forward. The dependence of

TAT on the frequency of T cell-APC encounters has been qualitatively described by

Grossman and Paul (1992, 2001), who discussed that antigen-stimulated expansion of T cells

is regulated through the combined action of T-cell tuning and of complementary cell-density

effects (Grossman and Paul, 2001; Grossman, 2004). Thus, Grossman and Paul (1992)

suggested that tuned T-cells would suppress other T cells by raising their activation

thresholds. Furthermore, they suggested (Grossman and Paul, 1992) that tuning would apply

to APCs as well, and hypothesized that the ability of APCs to stimulate T cells could be

down-regulated as the frequency of encounters with T cells increases. Therefore, our results

are in line with the general qualitative views of Grossman and colleagues.

3. Modeling tolerance mediated by regulatory CD4+CD25+ T lymphocytes

In recent years, a lot of information has been gathered on regulatory T cells within the

CD4+CD25+ pool (for a review see (Sakaguchi et al., 2001)). These cells seem to be

produced already differentiated in the thymus (Saoudi et al., 1996; Bensinger et al., 2001;

Jordan et al., 2001). They present unique transcription factors that confer them the regulatory

phenotype (Hori et al., 2003). Their expansion and persistence in the periphery is dependent

on recurrent interactions with APCs, which present self-antigens (Seddon and Mason, 1999;

Gavin et al., 2002). Regulatory T cells do not produce autocrine growth factors, namely IL-2

(Takahashi et al., 1998; Thornton and Shevach, 1998). Lafaille and colleagues (Furtado et al.,

2002) have provided evidence that in vivo regulatory T cell populations require IL-2 produced

by other cells. We have demonstrated on theoretical grounds that regulatory T cells must use

16/44

the T cells they suppress “as growth factor” (Leon et al., 2000; Leon et al., 2001). Regulatory

T cells may promote the differentiation of their targets into the regulatory phenotype, or use a

growth factor produced by their targets. We provided experimental evidence for this latter

possibility in vitro (Leon, 2002).

A MINIMAL MODEL

As in the previous section, our model follows the dynamics of a population of autoreactive T

cells whose activation, proliferation and survival depends on interactions with a homogeneous

population of APCs. This T cell population is made of two subpopulations of regulatory (TR)

and effector (TE) cells, with the same antigenic specificity. TE cells are responsible for

autoimmune disease if TR cells do not control their activation-dependent expansion. The

diagram in fig. 4A illustrates the basic processes in the model. Briefly, resting TR and TE cells

can die or form conjugates with free sites on the APCs. Conjugation can be productive,

resulting in T cell activation, or non-productive such that the T cell remains in resting state. T

cell activation is transient, and activated T cells will spontaneously rest. Activated TR and TE

cells, but not resting cells, will mutually interact. Activated TE cells, but not TR cells or resting

TE cells, produce a growth factor. The growth factor acts on the TE producing it in an

autocrine way, and on other activated TE and TR cells on a paracrine way. Activated TR cells

and TE cells divide as a function of this growth factor. The activation of a TE cell is inhibited

upon interaction with an activated TR cell. TR cells do not produce growth factors.

The mechanism underlying mutual interactions between TR and TE cells is a major

issue. Leon et al. (Leon et al., 2000; Leon et al., 2001; Leon et al., 2003; Leon et al., 2004)

have proposed and analysed a model in which mutual interactions between T cells require

their simultaneous conjugation with an APC, i.e. the formation of multicellular conjugates.

17/44

This mechanism is in accordance with the dependence of in vitro suppression on the ratio

between TR cell and APC numbers (Leon et al., 2001). However, Thornton and Shevach

(Thornton and Shevach, 2000) have shown that TR cells, which have been previously

activated by APCs, may suppress TE cells by direct cell-to-cell contact in vitro. The results

illustrated in the present article were obtained with a model assuming direct interactions

between activated TR and TE cells. When appropriate, we will pinpoint the differences with

the Leon et al. model. From the outset it is important to note that in the Leon et al. model,

efficient suppression of TE cells requires the presence of a minimum number of TR cells per

APC (Leon et al., 2001), while efficient suppression in the model used here merely requires a

minimum density of TR cells irrespective of the number of APCs. The qualitative differences

between the two models unfold from these different postulates about APC-dependence.

Assuming that the densities of conjugates, and of activated T cells are in quasi-steady

state, the reactional diagram in fig. 4 can be translated into the following pair of ordinary

differential equations (see derivation in Appendix C):

†

dRdt

= sEA RA -dR (11)

†

dEdt

= pEA -dE (12)

with:

†

RA =adRC

r + sEA

(13)

†

EA = -r(p + r) + dsa(RC - EC ) - 4d(p + r)rsaEC + -r(p + r) - dsa(RC - EC )( )2

2(p + r)s

(14)

†

RC =EA

dc + R + E

(15)

18/44

†

RC =RA

dc + R + E

(16)

where R and E are the total density of TR and TE cells, A is total density of APC-conjugation

sites (assumed to be constant), c is the conjugation rate, d is the deconjugation rate, d is the

death rate, s is the suppression rate, and r is the reversion rate of an activated T cell to the

resting state. This model is highly non-linear, and since we could not obtain closed

expressions for the steady states, we made numerical phase-plane and bifurcation analyses.

Like in the Leon et al. model (Leon et al., 2000), the richest phase-plane of this (R,E)

model has 4 steady states, and displays bistability (fig. 5A). The steady states are: the trivial

(0,0) state, corresponding to the extinction of TR and TE cells, which is unstable; an unstable

saddle-point where both TR and TE coexist, (R3,E3); a stable state of coexistence of TR and TE

cells, (R2,E2) ; and another stable state in which TR cells are competitively excluded by TE

cells, (0,E1). Following (Leon et al., 2000), we interpret the stable coexistence of TR and TE

cells as self-tolerance and the competitive exclusion of TR cells by TE cells as autoimmunity.

The (co-)existence of these steady states in the phase-plane is controlled by the relative

values of the parameters determining the net growth of the TR population (d, K, A, and s) , and

the net growth of the TE population (d, K, A, and p). Relative high net growth of the TR

population as compared to TE leads to a global stability of the self-tolerance state; while

relative low growth of TR cells results in disappearance of the self-tolerance state and global

stability of the autoimmunity state.

One important control parameter is the density of APCs, A. In fig. 5B we present a

typical bifurcation diagram of total density of TR and TE (R+E) as a function of the value of

A. Too low densities of APCs are unable to sustain any T cells in the population. The state

(0,0) is globally stable and the state (0,E1) is unstable and has no physical meaning because

E1<0. Following a transcritical bifurcation involving these two steady states, the state (0,E1)

becomes stable and also gains physical meaning (E1≥0) (a representative phase plane is

19/44

depicted in fig.5A-left). For an interval of relatively low values of A only TE cells can be

sustained in the population. For higher values of A, following a fold-bifurcation that brings in

the unstable saddle (R3, E3) and a stable state (R2,E2), the system becomes bistable, such that

depending on initial conditions either the autoimmunity or the tolerance states can be reached

(representative phase plane in fig.5A-middle). For even higher values of A there is another

transcritical bifurcation involving the unstable saddle (R3,E3) and the competitive exclusion

state (0,E1). The latter state becomes unstable and the previously unstable coexistence state

becomes stable but physically meaningless because R3 is now negative (i.e. the two non-trivial

nullclines intersect in the quadrant (R<0,E>0)). As a consequence of this bifurcation, the state

of coexistence of TE and TR cells (R2, E2) becomes the only stable state with physical

meaning. This is a major difference between the model presented here and the model of Leon

et al (Leon et al., 2000), where the increase of A never results in a bifurcation from the

bistability to the global stability regimes (fig.5C). In the present model, as the APC density

increases the size of the population at the coexistence steady state, E+R, tends asymptotically

to a constant value (suppression requires a minimal density of TR cells) (fig.5B). In the Leon

et al. model (Leon et al., 2000), E+R in the coexistence state increases linearly with the

density of APCs (suppression in the presence of more APCs requires more TR cells to “cover”

the same fraction of APCs) (fig.5C).

SPECIFIC DISCUSSION

The (R,E) model presented here and the model proposed and studied by Leon et al.

(Leon et al., 2000) offer a mechanism of self-tolerance by prevention of autoreactive T cell

expansion. These two models can readily explain the Sakaguchi phenomenon. In healthy

individuals the subpopulation of CD4+CD25- T cells is enriched in TE cells, while the

20/44

subpopulation of CD4+CD25+ T cells is enriched in TR cells. The fraction of TE cells is so

high in the first CD4 subset that its transfer into empty animals results in autoimmunity; while

the fraction of TR cells in the second CD4 subset is sufficiently high that when mixed with the

first subset leads to tolerance. Several authors (Annacker et al., 2001; Almeida et al., 2002;

Hori et al., 2002) have analysed the population dynamics of CD4+CD25+ and CD4+CD25- T

cells in recipient animals, showing that the presence of regulatory T cells reduces the apparent

steady state density of CD25- T cells. They also reported that when transferred alone

CD4+CD25+ T cells expand and persist in the recipients (Annacker et al., 2001; Almeida et

al., 2002; Hori et al., 2002), suggesting that this subset is an impure population of TR cells

containing also TE cells (which act as a source of growth factors), or that TR cells obtain

growth factors also from non-T cells. However, the number of CD25+ T cells recovered are

higher in the presence of CD25- (Demengeot, personal communication), confirming our

theoretical results according to which CD25- act as a source of growth factors, albeit non-T

cell derived growth factors might be also present.

The present (R,E) model retains several immunologically meaningful properties

previously identified in the Leon et al. model. For example, it predicts that diverse subclinic

infections will have a net protective effect against autoimmunity (Leon et al., 2004). However,

the Leon et al. model, but not the present (R,E) model, predicts a strong impact of changes in

APC density on the steady state attained within the bistability regime: a relatively fast

increase in APC density will force a switch from the self-tolerance to the autoimmunity state.

This switch from one stable state to another requires an APC-dependent suppressive

interaction, and is not recovered with the (R,E) model presented here, which features a direct

TR-TE interaction. This switch is biologically meaningful, since it provides a common

rationale for the etiology of some autoimmune diseases that are associated to specific

infections (Leon et al., 2004), for the ability to cause experimental autoimmune pathologies

21/44

through immunization with self-antigens in adjuvants (Leon et al., 2000), and for the ability to

experimentally induce autoimmunity by neonatal thymectormy (Leon et al., 2000; Leon et al.,

2003). None of these properties are recovered in the simple (R,E) model presented here.

Therefore we conclude that the Leon et al. model might be capturing better the reality of self-

tolerance mediated by regulatory T cells in vivo.

4. General Discussion

This article reviewed mathematical models of self-tolerance by control of expansion of

autoreactive T cell populations mediated by two mechanisms: tunable activation thresholds

without suppression or suppression by regulatory T cells without tuning. We have shown that

proliferative anergy in our simple TAT model decreases with T cell density relative to APCs.

Due to this property, the model can explain efficient control of expansion of autoreactive T

cells, but not their persistence. In the second model, the existence of a tolerance steady-state in

which TR and TE cells coexist is compatible with the fact that from every self-tolerant

individual autoreactive T cells can be purified that can cause autoimmunity. In contrast with

the simple TAT model analyzed here, this will be true even in the absence of a continuous

influx of cells from the thymus. Extrapolating from these examples, it is clear that models that

can explain the equilibrium between cell proliferation and death, in the absence of external

sources, must include some form of T cell-density dependent suppression. As noted earlier,

some conceptual models have postulated an interplay between suppression and tuning of

activation thresholds (Grossman and Paul, 2001; Grossman, 2004), but the involvement of the

later regulatory mechanism remains to be established. Our results indicate that suppression

mediated by regulatory T cells is sufficient to explain the prevention of pathologic

autoimmune responses by effector T cells in the Sakaguchi phenomenon. Analysis of

22/44

mathematical models of suppression by regulatory T cells suggested that the persistence and

growth of the regulatory population is dependent on the autoreactive effector T cells they

control, and that this dependency will increase the efficiency of suppressive function. This

crosstalk between regulatory and effector cells can fully account for adoptive transfers of

tolerance by CD4+CD25+ T cells, as well as for several other features of tolerance. In the

following, we extend the discussion, considering more generally the problem of self-tolerance

regulation and also the related issue of adaptation of the immune system to chronic antigen

stimuli.

TUNING OF ACTIVATION THRESHOLDS AND SUPPRESSION BY REGULATORY T CELLS

Tunable activation thresholds and suppression by regulatory T cells are not mutually

exclusive mechanisms of self-tolerance. Already in their original proposal, Grossman & Paul

(Grossman and Paul, 1992) suggested that anergic cells, with higher activation thresholds,

could render naive cells anergic. Suppression by anergic cells has been shown in vitro

(Lombardi et al., 1994; Taams et al., 1998), albeit the results are controversial (Kuniyasu et

al., 2000), and the suppressive mechanism is not defined.

What dynamic properties are expected if T cell anergy is induced and maintained both

by interactions with APCs and by interactions with other anergic T cells? We earlier studied

(Leon et al., 2000) a mathematical model in which TR cells convert TE cells into the regulatory

phenotype, showing that it has the same properties of a model in which TR cells receive a

growth factor from TE cells. What would tunable activation threshold bring in addition to this?

Consider the dependency of the fraction a of TE and TR cells activated upon conjugation with

APCs on the frequency of conjugations and tuning. Consider also the frequency of T cell-APC

interactions at the stable steady states of the (R,E) system with fixed a. Essentially the phase

23/44

plane of the system will be maintained if the parameters are such that conjugations with the

APCs at the stable steady states are rare enough such that threshold tuning is not significant,

i.e. a is practically constant; under these conditions, one would expect the extinction of both

effector or regulatory T cells to be stable, instead of unstable as in fig. 5C. However, if the

parameters are such that the interactions with APCs are frequent enough to reduce the fraction

of conjugated TE and TR cells that become activated then the steady states may disappear.

These additional complications of coupling suppression and anergy induction are non-trivial

and require proper modeling. Grossman and Paul (1992, 2001) have discussed these issues

extensively and, although they do not provide explicit mathematical models, their “conceptual

models” would be a good stepping-stone.

ADAPTATION IN THE IMMUNE SYSTEM: CELLULAR OR POPULATIONAL?

Several lines of evidence indicate that the immune system shows adaptation to

continuous antigenic stimuli. Typically, chronically stimulated T cell populations are shown

to become unresponsive when tested as a bulk (e.g. Tanchot et al. (2001); Singh and Schwartz

(2003)). Acquisition of this bulk unresponsiveness is often interpreted as the adaptation of

individual cells by raising their activation thresholds, however, this interpretation is not

unique. Indeed, Grossman and colleagues (1992, 2001) have argued that adaptation could

happen at the level of the population as well as at the level of the individual cell signaling

machinery. The mathematical analysis described here well illustrates this argument, since the

(R,E) models show adaptation of the populations to chronic stimuli much in the same way as

the kinase and phosphatase in the individual cell model. Thus, in the (R,E) models, when the

system is in the tolerance state (be it within the bistable or the globally stable regimes), a

sudden increase in antigen-bearing APCs to a new set point will trigger a transient response

24/44

corresponding to the orbit of the system attaining the new steady state. This will be

characterised by a transient expansion of TE cells that will be eventually controlled by TR

cells. In the Leon et al. model (Leon et al., 2004), within the bistability regime, a fast increase

of APC density may force a tolerance state to switch to an autoimmunity state.

Given the above considerations, the question is to what extent the adaptation scored as

acquisition of bulk T-cell unresponsiveness happens at the level of single cell signaling or at

the population level? Answering this question experimentally is not straightforward. For

example, to experimentally rule out that an interaction between T cells regulates a response

requires the use of limiting dilution analysis or single cell analysis, where the interactions are

greatly disfavored or simply prevented (Dozmorov et al., 2000). Only if the frequency of

responder T cells is not affected by diluting away T cell interactions can one definitively

conclude that there was single cell adaptation, and eventually quantify the extent of induction

of single cell unresponsiveness. These assays are, however, rarely performed in assessing

adaptation of bulk T cell responses, which prevents an unequivocal interpretation of the

results.

In this article we used two simple models to gain insight into the tolerance in the

Sakaguchi phenomenon. Can these simple models also be used to gain insight into the

mechanism underlying the acquisition of bulk T cell unresponsiveness? We believe so. As we

have seen, the T cell-density dependencies of unresponsiveness by suppression and by TAT-

dependent single cell anergy are opposite to each other. Suppression increases as the density

of TR cells per APC increases, and bulk T cell responsiveness (i.e. the response of a mixture of

TR and TE cells) should decrease accordingly. In contrast, the average activation threshold, in

our model the activity of the inhibitory phosphatase, is tuned down when the ratio of T cells

per APC increases (fig. 3B), and therefore bulk T cell responsiveness should increase

accordingly. This offers a sort of “rule of thumb” for assessing what might be the predominant

25/44

mechanism of adaptation in a given experimental setting in which the bulk responsiveness can

be measured as a function of T-cell density per APC.

HOW CAN EFFICIENT RESPONSES TO FOREIGN ANTIGENS AND ROBUST SELF-TOLERANCE

COEXIST IN THE IMMUNE SYSTEM?

The most important question about any self-tolerance mechanism is: how can efficient

immune responses to foreign pathogens be mounted, while the immune system remains

robustly self-tolerant?

The solution to this puzzle under the general TAT framework is that immune

responses will be mounted to any antigen, self or foreign, whose presentation on the APCs

increases suddenly (Grossman and Paul, 2000). Using a mathematical model, Scherer et al.

(2004) have shown that raising the activation thresholds of autoreactive T cells in the thymus,

as posited before by Grossman and Singer (1996), would be a more efficient way of ensuring

efficient self-nonself discrimination than classical deletional mechanisms. Based on a TAT

model, featuring also a TCR-dependent kinase-phosphatase cycle, Vand De Berg and Rand

(2004) have argued that tuning would render T cells uniform across repertoire and space, in

terms of their capacity to respond to foreign antigens, and that pre-tuning in thymus would

facilitate tolerance to self-antigens in the periphery. We reached a similar conclusion using

our Monte-Carlo simulations (Sousa, 2003). Furthermore, the individual cell responses to

foreign antigens would be facilitated and more sustained if the increase in the magnitude of

the stimulus per APC (s) is not concomitant with a large increase of stimulatory APCs.

Hence, an increase in APCs, which is often associated with infections, will increase the

frequency of conjugation events and therefore facilitates adaptation. This facilitation of

adaptation might be counteracted in vivo by the fact that once the T cells are activated they

26/44

lower their thresholds of activation, and perhaps become more resistent to tuning (Grossman

and Paul, 2001; Iezzi et al., 1998).

Regarding tolerance mediated by regulatory T cells, one of the aspects of the question

above is that foreign antigens are always co-presented with self-antigens, and therefore

autoreactive T cells could prevent immune responses (Leon et al., 2003). Based on simulation

results, we have argued (Leon et al., 2003) that immune responses can be efficiently elicited

to those foreign antigens that displace sufficient self-antigens from the APCs and/or that are

presented concomitantly with a marked increase in APCs. Another perhaps complementary

solution, suggested by the typical bifurcation diagrams of the (R,E) models (fig.5B and C), is

that the repertoire of regulatory T cells would be strongly biased towards self-antigens.

Consider a scenario in which most T cell clones in circulation recognize too few APCs to

sustain regulatory T cells. This is not unlikely given the fact that thymic deletion eliminates

those T cells that would respond strongly to ubiquitous antigens. These T cell clones will

contain only TE cells but they would not cause autoimmunity because their expansion is

limited by too few available APCs. Rarer T cell clones will recognize enough APCs such that

they could expand to very high numbers, and thus could cause autoimmunity. In this case,

however, APC-density is sufficient to sustain TR cells, and thus clonal expansion is controlled.

Although within this bistability regimen the autoreactive clones can reach either

autoimmunity or tolerance, robust tolerance to these antigens will follow if the thymus exports

enough TR cells to ensure that any (R, E) population will be seeded within the basin of

attraction of the state of TR-TE coexistence (Leon et al., 2003). In this scenario, our model

predicts that the T cell repertoire can be divided into two sets of lymphocyte clones: a larger,

more diverse set of small clones containing only TE cells, and a less diverse set of small

clones, containing both TE and TR cells. In the first set, clonal sizes are determined only by

APC availability, while in the second set clonal sizes are determined by suppression mediated

27/44

by regulatory T cells. The dynamics of the first set would be that of the competition system

modeled by De Boer and Perelson (De Boer and Perelson, 1994; De Boer and Perelson,

1997). The dynamics of the second set would be that of the system studied by Leon et al.

(Leon et al., 2000; Leon et al., 2003). In this scenario, immune responses driven mainly by an

increase in APCs would be obtained from the first set of clones, while tolerance to self would

be ensured by the second set of clones. The plausibility of this scenario depends critically on

TCR crossreactivity and copresentation of peptides on the same APCs: whether a foreign

antigen will elicit an immune response will depend on how many clones from the first and the

second set will recognize peptides on the same APCs (Leon et al., 2003). The constraints on

repertoire size and crossreactivity/copresentation necessary for efficient self-nonself

discrimination have been studied under the assumption that tolerance is mediated by clonal

deletion (De Boer and Perelson, 1993; Faro et al., 2004, and references therein) and more

recently by tuning (Scherer et al. 2004; Van de Berg, 2004). The scenario we propose offers

also another type of constraint on repertoire sizes and crossreactivity that could be amenable

to similar modeling studies.

5. Concluding Remarks

Hitherto the mechanisms of self-tolerance are essentially unresolved. We have used

mathematical models to gain insights into these mechanisms. The models were designed as

simple as possible in order to allow a better understanding of their knots and bolts. Therefore,

while evidently unrealistic, such models may provide clues on how to make them more

realistic. Despite their conceptual simplicity the models are highly non-linear, requiring non-

trivial analysis. Model analysis was based mainly on simple phase-plane and bifurcation

analysis, which can be related to biology in straightforward, generic ways. We bootstrap the

lack of analytic solutions through quasi-steady state approximations, graphical

28/44

representations, and numerical solutions. Our conclusions are grounded on worked examples

from other fields notably from statistical mechanics, and population dynamics. More than

discussing the details of the mathematical derivations, which can be found in other

publications, we have discussed and compared the assumptions and interpretations of different

models. We believe that such continued critical discussion is instrumental in uncovering the

basic rules of the immunological game, by producing more realistic models.

29/44

Acknowledgements

The authors are greatful to Jocelyne Demengeot, Zvi Grossman, António Coutinho, António

Bandeira, Nuno Sepúlveda, and Íris Caramalho for many, many discussions on self-tolerance,

and for their encouragement. This article is based on the Ph.D. thesis of Kalet Leon and João

Sousa (available online at: http://eao.igc.gulbenkian.pt/ti/index.html). The work was

finantially supported by Fundação para a Ciência e Tecnologia: grants P/BIA/10094/1998,

POCTI/36413/99, and POCTI/MGI/46477/2002; and fellowships to JF

(Praxis/BCC/18972/98), JS (BD/13546/97), KL (SFRH/BPD/11575/2002), DM

(SFRH/BD/2960/2000) and TP (SFRH/BD/10550/2002).

30/44

References

Acha-Orbea, H. and H. R. MacDonald (1995). "Superantigens of mouse mammary tumor

virus." Annu Rev Immunol 13: 459-86.Almeida, A. R., N. Legrand, M. Papiernik and A. A. Freitas (2002). "Homeostasis of

peripheral CD4+ T cells: IL-2R alpha and IL-2 shape a population of regulatory cellsthat controls CD4+ T cell numbers." J Immunol 169(9): 4850-60.

Annacker, O., R. Pimenta-Araujo, et al. (2001). "CD25+ CD4+ T cells regulate the expansion

of peripheral CD4 T cells through the production of IL-10." J Immunol 166(5): 3008-18.

Barthlott, T., G. Kassiotis and B. Stockinger (2003). "T cell regulation as a side effect ofhomeostasis and competition." J Exp Med 197(4): 451-60.

Bensinger, S. J., A. Bandeira, et al. (2001). "Major histocompatibility complex class II-

positive cortical epithelium mediates the selection of CD4(+)25(+) immunoregulatoryT cells." J Exp Med 194(4): 427-38.

Burnet, F. (1957). "A modification of Jerne's theory of antibody production using the conceptof clonal selection." Aust J Sci 20: 67-69.

Clements, J. L., N. J. Boerth, J. R. Lee and G. A. Koretzky (1999). "Integration of T cell

receptor-dependent signaling pathways by adapter proteins." Annu Rev Immunol 17:89-108.

De Boer, R. J. and A. S. Perelson (1993). "How diverse should the immune system be?" Proc

R Soc Lond B 252(1335):171-5.De Boer, R. J. and A. S. Perelson (1994). "T cell repertoires and competitive exclusion." J

Theor Biol 169(4): 375-90.De Boer, R. J. and A. S. Perelson (1997). "Competitive control of the self-renewing T cell

repertoire." Int Immunol 9(5): 779-90.

Dosmorov, I, M.D. Eisenbraun and I. Lefkovitz (2002). "Limiting dilution analysis: from

frequencies to cellular interactions". Immunol Today. 21(1):15-8.

Faro, J. S.Velasco, A Gonzalez-Fernandez and A. Bandeira (2004)." The impact of thymic

antigen diversity on the size of the selected T cell repertoire." J Immunol. 2004 Feb

15;172(4):2247-55

31/44

Furtado, G. C., M. A. Curotto de Lafaille, N. Kutchukhidze and J. J. Lafaille (2002).

"Interleukin 2 signaling is required for CD4(+) regulatory T cell function." J Exp Med196(6): 851-7.

Gavin, M. A., S. R. Clarke, et al. (2002). "Homeostasis and anergy of CD4(+)CD25(+)suppressor T cells in vivo." Nat Immunol 3(1): 33-41.

Germain, R. N. and I. Stefanova (1999). "The dynamics of T cell receptor signaling: complex

orchestration and the key roles of tempo and cooperation." Annu Rev Immunol 17:467-522.

Grossman Z., B. Min, M. Meier-Schellersheim M, and W.E. Paul. (2004) Concomitant

regulation of T-cell activation and homeostasis. Nat. Rev. Immunol. 4(5):387-95.

Grossman, Z. and W. E. Paul (1992). "Adaptive cellular interactions in the immune system:

the tunable activation threshold and the significance of subthreshold responses." Proc

Natl Acad Sci U S A 89(21): 10365-9.Grossman, Z. and W. E. Paul (2000). "Self-tolerance: context dependent tuning of T cell

antigen recognition." Semin Immunol 12(3): 197-203; discussion 257-344.Grossman, Z. and W. E. Paul (2001). "Autoreactivity, dynamic tuning and selectivity." Curr

Opin Immunol 13(6): 687-98.

Grossman, Z. and A. Singer (1996). "Tuning of activation thresholds explains flexibility in the

selection and development of T cells in the thymus" Proc Natl Acad Sci U S A 93(25):

14747-52.

Hori, S., T. L. Carvalho and J. Demengeot (2002). "CD25+CD4+ regulatory T cells suppress

CD4+ T cell-mediated pulmonary hyperinflammation driven by Pneumocystis cariniiin immunodeficient mice." Eur J Immunol 32(5): 1282-91.

Hori, S., T. Nomura and S. Sakaguchi (2003). "Control of regulatory T cell development bythe transcription factor Foxp3." Science 299(5609): 1057-61.

Iezzi, G., K.Karjalainen, and A. Lanzavecchia (1998) The duration of antigenic stimulation

determines the fate of naive and effector T cells. Immunity 8(1): 89-95.Jordan, M. S., A. Boesteanu, et al. (2001). "Thymic selection of CD4+CD25+ regulatory T

cells induced by an agonist self-peptide." Nat Immunol 2(4): 301-6.Kisielow, P., H. Bluthmann, et al. (1988). "Tolerance in T-cell-receptor transgenic mice

involves deletion of nonmature CD4+8+ thymocytes." Nature 333(6175): 742-6.

Koshland, D. E., Jr., A. Goldbeter and J. B. Stock (1982). "Amplification and adaptation inregulatory and sensory systems." Science 217(4556): 220-5.

32/44

Kuniyasu, Y., T. Takahashi, et al. (2000). "Naturally anergic and suppressive CD25(+)CD4(+)

T cells as a functionally and phenotypically distinct immunoregulatory T cellsubpopulation." Int Immunol 12(8): 1145-55.

Langman, R. E. and M. Cohn (1987). "The E-T (elephant-tadpole) paradox necessitates theconcept of a unit of B-cell function: the protection." Mol Immunol 24(7): 675-97.

Lederberg, J. (1959). "Genes and antibodies." Science 129(3364): 1649-53.

Leon, K. (2002). A quantitative approach to dominant tolerance. PhD Dissertation. Porto,University of Porto pp. 156.

Leon, K., J. Faro, A. Lage and J. Carneiro (2004). "Inverse correlation between the incidencesof autoimmune disease and infection predicted by a model of T cell mediated

tolerance." J Autoimmun 22(1): 31-42.

Leon, K., A. Lage and J. Carneiro (2003). "Tolerance and immunity in a mathematical modelof T-cell mediated suppression." J Theor Biol 225(1): 107-26.

Leon, K., R. Perez, A. Lage and J. Carneiro (2000). "Modelling T-cell-mediated suppression

dependent on interactions in multicellular conjugates." J Theor Biol 207(2): 231-54.Leon, K., R. Perez, A. Lage and J. Carneiro (2001). "Three-cell interactions in T cell-

mediated suppression? A mathematical analysis of its quantitative implications." JImmunol 166(9): 5356-65.

Lombardi, G., S. Sidhu, R. Batchelor and R. Lechler (1994). "Anergic T cells as suppressor

cells in vitro." Science 264(5165): 1587-9.Levchenko, A. and P.A. Iglesias. (2002). "Models of Eukaryotic Gradient Sensing:

Application to chemotaxis of amoeba and neutrophils". Biophysic. J. 82 (1): 50-63.Milutinovic, D., J. Carneiro, M. Athans and P. Lima (2004). "Modeling the hybrid dynamics

of TCR signaling in immunological synapses." J Theor Biol submitted.

Nicholson, L. B., A. C. Anderson and V. K. Kuchroo (2000). "Tuning T cell activationthreshold and effector function with cross-reactive peptide ligands." Int Immunol

12(2): 205-13.Pereira, P., L. Forni, et al. (1986). "Autonomous activation of B and T cells in antigen-free

mice." Eur J Immunol 16(6): 685-8.

Polic, B., D. Kunkel, A. Scheffold and K. Rajewsky (2001). "How alpha beta T cells deal withinduced TCR alpha ablation." Proc Natl Acad Sci U S A 98(15): 8744-9.

Sakaguchi, S., N. Sakaguchi, et al. (1995). "Immunologic self-tolerance maintained byactivated T cells expressing IL-2 receptor alpha-chains (CD25). Breakdown of a single

33/44

mechanism of self-tolerance causes various autoimmune diseases." J Immunol 155(3):

1151-64.Sakaguchi, S., N. Sakaguchi, et al. (2001). "Immunologic tolerance maintained by CD25+

CD4+ regulatory T cells: their common role in controlling autoimmunity, tumorimmunity, and transplantation tolerance." Immunol Rev 182: 18-32.

Saoudi, A., B. Seddon, et al. (1996). "The physiological role of regulatory T cells in the

prevention of autoimmunity: the function of the thymus in the generation of theregulatory T cell subset." Immunol Rev 149: 195-216.

Scherer, A., A. Noest and R.J. de Boer (2004) "Activation-threshold tuning in an affinity

model for the T-cell repertoire" Proc. R. Soc. Lond. B 271(1539):609-16.

Schwartz, R.H. (1990). "A cell culture model for T cell clonal anergy." Science 248: 1349.

Seddon, B. and D. Mason (1999). "Peripheral autoantigen induces regulatory T cells thatprevent autoimmunity." J Exp Med 189(5): 877-82.

Singh, N. J. and R. H. Schwartz (2003). "The strength of persistent antigenic stimulation

modulates adaptive tolerance in peripheral CD4+ T cells." J Exp Med 198(7): 1107-17.

Smith, K., B. Seddon, et al. (2001). "Sensory adaptation in naive peripheral CD4 T cells." JExp Med 194(9): 1253-61.

Sousa, J. (2003). Modeling the antigen and cytokine receptors signalling processes and their

propagation to lymphocyte population dynamics. PhD Dissertation. Lisbon,University of Lisbon. pp. 161.

Stefanova, I., B. Hemmer, et al. (2003). "TCR ligand discrimination is enforced by competingERK positive and SHP-1 negative feedback pathways." Nat Immunol 4(3): 248-54.

Taams, L. S., A. J. van Rensen, et al. (1998). "Anergic T cells actively suppress T cell

responses via the antigen-presenting cell." Eur J Immunol 28(9): 2902-12.Takahashi, T., Y. Kuniyasu, et al. (1998). "Immunologic self-tolerance maintained by

CD25+CD4+ naturally anergic and suppressive T cells: induction of autoimmune

disease by breaking their anergic/suppressive state." Int Immunol 10(12): 1969-80.Tanchot, C., D. L. Barber, L. Chiodetti and R. H. Schwartz (2001). "Adaptive tolerance of

CD4+ T cells in vivo: multiple thresholds in response to a constant level of antigenpresentation." J Immunol 167(4): 2030-9.

34/44

Thornton, A. M. and E. M. Shevach (1998). "CD4+CD25+ immunoregulatory T cells

suppress polyclonal T cell activation in vitro by inhibiting interleukin 2 production." JExp Med 188(2): 287-96.

Thornton, A. M. and E. M. Shevach (2000). "Suppressor effector function of CD4+CD25+immunoregulatory T cells is antigen nonspecific." J Immunol 164(1): 183-90.

Van den Berg, H and D.A. Rand (2004) "Dynamics of T cell activation threshold tuning" J.

Theor. Biol. 228 (3): 397-416.Witherden, D., N. van Oers, et al. (2000). "Tetracycline-controllable selection of CD4(+) T

cells: half-life and survival signals in the absence of major histocompatibility complexclass II molecules." J Exp Med 191(2): 355-64.

Wong, P., G. M. Barton, K. A. Forbush and A. Y. Rudensky (2001). "Dynamic tuning of T

cell reactivity by self-peptide-major histocompatibility complex ligands." J Exp Med193(10): 1179-87.

35/44

Figure Legends

Figure 1 – Illustration of the experiments of Sakaguchi et al. demonstrating the existence of

autoreactive T cells in healthy individuals. Purified CD4+CD25- T cells, but not

CD4+CD25+T cells, from healthy animals will cause autoimmune diseases in recipient

animals. An interaction between the two subsets of CD4 cells prevents disease development in

the same recipients.

Figure 2 – Illustration of a model of the population dynamics of lymphocytes with tunable

activation thresholds indicating the cellular processes (A) and the molecular processes of the

cell machinery (B).

Figure 3 – Analysis of the model of the population dynamics of T cells with tunable activation

thresholds. A- Kinetics of the phosphatase (black line) and the kinase (gray line) which are

downstream of TCR stimulus in an individual T cell. The probability that the phosphatase

activity P supersedes the kinase activity K at the instant of deconjugation increases with the

frequency of encounters. (Top: c(A-C)=0.20; bottom: c(A-C)=0.048). B– Stationary

probability density function of the phosphatase activity P in the population of conjugated T

cells at the indicated densities (P0 = 60; s = 1000). C- Phase diagram of the model indicating

the death rate (dashed line) and growth rates (solid lines) for the indicated values of the

control parameter P0/K0; the dots indicate the stable (black) and unstable (white) steady states

for the reference value P0/K0=60. D- Bifurcation diagram obtained by varying the control

parameter P0/K0 that determines the adaptation capacity of the signaling machinery; Stable

and unstable steady states are indicated by solid and dashed lines respectively. (Reference

36/44

parameters: rP=0.027 day-1; K0=1 au; P0=60 au; s=1000; A=8 cells; d=6 day-1; c=0.06 cell-

1day-1; d=0.02 day-1; au=relative activity units).

Figure 4 — Illustration of the regulatory T cell population model indicating all the processes

in which APCs, TE and TR cells are involved.

Figure 5 – Phase planes for the regulatory T cell population model, and bifurcation as a

function of the APC density. A and B- The model presented in the main text was used with

parameters: p=2 day-1, r =0.3 day-1, d=6 day-1; c=0.06 cdu-1day-1; d=0.02 day-1; a =1 and

s=0.07 cdu-1day-1; cdu=relative cell density units. The three phase diagrams in A were

obtained from left to right with A= 5, 10 and 15 cdu respectively. C- Bifurcation diagram for

the model described by Leon et al. (2003).

37/44

Appendix A. Derivation of the quasi-steady state model of a single T-cell population

The diagram in fig.1A can be translated into the following two differential equations and one

conservation equation:

†

d TF

dt= (1+ a)dC -cTF AF - d ⋅TF

(A1)

†

dCdt

= cTF AF - dC (A2)

†

A = AF + C (A3)

where TF is the density of free T cells, TC is the density of APC-T-cell conjugates, AF is the

density of free APCs and A is the total density of APCs. The parameters are the rate constant

of conjugate formation c , the rate constant of conjugate dissociation d , and the death rate

constant d. a is the probability that a T-cell is activated following activation. It depends on the

internal state of the T-cell and it is defined according to eqn.10 in the main text, which uses

the stationary probality density function of the phosphatase activity in the conjugated derived

in Appendix B.

We are interested in following the total density of T cells in time denoted T:

†

T = TF + C (A4)

One practical reason to do this is that with the available experimental techniques it is very

difficult to count free and conjugated T cells in vivo. Instead experimentalists isolated the

mixture of T cells (conjugated or free) and counted them.

Taking the derivative of both sides of eqn. A4 we obtain:

38/44

†

d Tdt

=dTF

dt+

dCdt

= daC -dTF = daC -d(T - C) (A5)

which is to eqn.1 in the main text.

Assuming that the conjugates are in quasi-steady state we have:

†

dCdt

= cTF AF - dC = 0 (A6)

Substituting TF and AF by their expression in terms of A, T and TC we obtain a second order

equation:

†

c(T - C)(A - C) - dC = 0 (A6)

Solving it we obtain two solutions, a negative and a positive. Only the positive solution is

physically meaningful and thus was considered and corresponds to eqn. 2 in the main text.

For phase-space and bifurcation analyses of this one-dimensional model we used the software

Mathematica. The steady states were calculated numerically for each combination of

parameters using the FindRoot routine of Mathematica, which implements the Newton

method. The stability of each of these solutions was determined by linear stability analysis.

Appendix B. Derivation of the stationary distribution of phosphatase activity in a population

of T cells

The dynamics of probability density functions (PDFs) of the phosphatase activity in the

subpopulations of conjugated and free T cells, respectively rC and rF, are described by the

39/44

following set of first order partial differential equations:

†

∂rC

∂t+

∂∂P

(PC rC ) = -drC + cE rF (B1)

†

∂rF

∂t+

∂∂P

(PF rF ) = drC - cE rF (B2)

where PC and P F are the functions governing the dynamics of the phosphatase in the

conjugated and free regimes (i.e. the right hand side of eqn. 4 with s>0 and s= 0

respectively):

†

PC = rP P0(1+ s) - P( ) (B3)

†

PF = rP P0 - P( ) (B4)

and cE=cAF=c(A-C) and d are the per T cell transition rates from the free to the conjugated

state and from the conjugated to the free state, respectively.

In search for the steady state solutions we make

†

∂PC

∂t= 0 , ∂PF

∂t= 0

Ê

Ë Á

ˆ

¯ ˜ and obtain the

following set of ordinary differential equations:

†

∂∂P

(PC rC ) = -drC + cE rF (B5)

†

∂∂P

(PF rF ) = drC - cE rF (B6)

Noticing that the right hand sides of these two equations are symmetrical we can add them

obtaining the following conservation:

†

∂∂P

(PC rC + PF rF ) = 0 (B7)

which upon integration leads to:

†

PC rC + PF rF = K (B8)

Because

†

rC and

†

rF are PDFs, which cannot be negative, there must be at least one value P1

such that

†

rC (P1) = rF (P1) = 0 . Therefore we have K=0 which leads to the following relation:

40/44

†

PF rF = -PC rC (B9)

Solving this equation for

†

rF and substituting in eqn. B2 we obtain the following ordinary

differential equation:

†

∂rC

∂P= -

dPC

+cE

PF

-∂PC

∂PÊ

Ë Á

ˆ

¯ ˜ rC (B10)

The solution of this equation is:

†

rC = Ne-

dPC

+cEPF

-∂PC∂P

1PC

dPÚ (B11)

Given the definitions of PF and PC according to eqns B3 and B4 the integrals are:

†

-d

PCdPÚ = log P0 (1+s )-P

drP (B12)

†

cEPF

dPÚ = log P0 -P-

cErP (B13)

†

-∂PC∂P

1PC

dPÚ = log P0 (1+s )-P-1 (B14)

Replacing these integrals in eqn. B11 we get the following equation that corresponds to eqn. 9

in the main text.

†

rC = N P0(1+ s ) - PdrP

-1 P0 - PcErP , P0 £ P £ P0(1+ s )

0 , otherwise

Ï Ì Ô

Ó Ô (B15)

The solution is branched because at the steady state the values of P are always contained in

the interval [P0,P0(1+s)], whose extremes are the steady state values of P predicted according

to eqn 4 in the main text if T cells would be either always free or always conjugated,

respectively.

Appendix C. Derivation of quasi-steady state model of T-cell population containing TR and

TE cells.

41/44

The diagram in fig.4 can be translated into the following set of differential equations and one

conservation equation:

†

d RF

dt= 2sRA EA + rRA + d(1-a)RC -cRF AF - d ⋅RF

(C1)

†

d EF

dt= sRA EA + 2 pEA + rEA + d(1-a)EC -cEF AF - d ⋅ EF

(C2)

†

d RC

dt= -dRC +cRF AF

(C3)

†

d EC

dt= -dEC + cEF AF

(C4)

†

d RA

dt= daRC - sRA EA - rRA

(C5)

†

d EA

dt= daEC - sRA EA - pEA - rEA

(C6)

†

A = AF + RC + EC

(C7)

where RF (or EF) is the density of free TR (or TE) cells, RC (EC) is the density of conjugated TR

(TE) cells, RA (EA) is the density of activated TR (TE) cells, AF is the density of free APCs, and

A is the total density of APCs. The parameters are the rate constant of conjugate formation c ,

the rate constant of conjugate dissociation d , the constant rate of reversion of the activated

state to the resting state r, the constant rate of division of activated TE cells that give rise to

two resting TE cells p, the constant rate of suppression s which leads to the reversion of

activated TE cells to the resting state without division but will lead to division of activated TR

cells into two resting TR cells , and the death rate constant d. a is the probability that a T-cell

is activated following activation. Biologically, it is the same a as in the model in appendix A.

In the model of tuning we made it dependent on the internal state of the cell and calculated it

according the PDF of P, but here it is a simple constant.

42/44

As before (Appendix A), we are interested in the dynamics of the total densities of TR and TE ,

denoted R and E respectively. The respective derivatives are:

†

dRdt

=dRF

dt+

dRC

dt+

dRA

dt= sEA RA -dRF

(C8)

†

dEdt

=dEF

dt+

dEC

dt+

dEA

dt= pEA -dEF

(C9)

For maximum simplicity, we assume that the densities of conjugated and activated T cells are

negligible when compared to the total densities at any time point. Thus, we have:

†

R = RF + RC + RA ª RF , E = EF + EC + EA ª EF (C10)

These approximations (C10) are valid as long as c<<d and ad≤r, which ensure that at

equilibrium the density of conjugated cells will be much smaller than the density of free cells,

and the density of activated cells is at maximum identical to the density of conjugated cells.

Under these assumptions, equations C8 and C9 become respectively equations 11 and 12 in

the main text.

We assume that the conjugate densities are in quasi-steady, obtaining the following

expressions for RC and EC:

†

RC =cd

AF RF , EC =cd

AF EF (C11)

43/44

Substituting these expressions in the conservation equation APCs (eqn C7) and solving to AF

we obtain:

†

AF =A

1+cd

RF +cd

EF (C12)

Substituing back in eqn. C8 we obtain the following expressions for the conjugates of TR and

TE cells:

†

RC =ARF

dc

+ RF + EF

, EC =AEF

dc

+ RF + EF

(C13)

which leads to eqns. 15 and 16 in the main text following the approximations in eqn C10.

We assume also that the activated T cells are also in quasi-steady state. Setting dRA/dt=0 and

solving eqn. C5 in order to RA we obtain:

†

RA =adRC

r + sEA

(C14)

Setting dEA/dt=0 in equation C6 and substituting RA according to eqn C14 yields the

following second order equation in EA:

†

0 = rdaEC + sdaEC - sadRC - r(p + r)( )EA - s(p + r)(EA )2 (C15)

Only one of the two solutions is positive and thus biologically meaningful being presented as

eqn. 14 in the main text.

44/44

For phase-space and bifurcation analyses of this two dimensional (R,E) model we used the

software Mathematica. Closed-form expressions were obtained for the nullclines, which are

parabolic curves in the plane (R,E), albeit no closed expressions could be obtained for the

non-trivial steady states. For each parameter set, the steady states were obtained numerically

by finding all the intersections between the pairs of E and R nullclines, i.e. by finding the

zeros of the subtraction of the two corresponding nullclines using the Newton method

(implemented in the function FindRoot of the software). Linear stability analysis of each

steady state was performed identifying stable and unstable states. The phase planes

corresponding to a particular parameter set were drawn by plotting the nullclines and the

stable and unstable steady states (represented as filled and empty circles respectively) in the

physically meaningful quadrant (both variables are null or positive). The bifurcation diagram

as a function of parameter A was obtained by plotting the sum of the state variables (R+E) in

each physically meaningful steady state. Stable states are represented with continuous lines

and unstable states with dashed lines.

CD4+CD25+ T cells

CD4+CD25- T cells

healthy

healthy

autoimmunedisease

Figure 1

c

d(1-a) da

d

( )

W W

K

P

s

B

AActivated T cell

Resting T cell

APC

Legend

W

W Inactive adapter

Active adapter

KP Inhibitory phosphatase

Activatory kinase

s TCR-dependent stimulus

rKK0(1+s) rk

rPP0(1+s) rP

Figure 2

Time (day x10-2)

10 2 3 4 5

100

1000

10000

10

100

1000

10000

10

A

C

T cell density, T

0

0.0002

0.0004

0.0006

120

60

1

0 500 1000 1500 2000 2500

B

Adaptation capacity,P0/K0

D

0 20 40 60 80 100 1200

500

1000

1500

2000

2500

Log10(P)

2.0 2.5 3.0 3.5 4.0 4.5

Figure 3

c

d(1-a) da

c

d(1-a) da

r

r

p

++ s

d

d

Activated TE cell

Resting TE cell

APC

Legend

Resting TR cell

Activated TR cell

Figure 4

TR cell density, R

A

B

APC density, A0 5 10 15 20

0

1000

2000

3000

4000

APC density, A

C

0 1000500 1500 0 1000500 1500 0 1000500 15000

1000

2000

3000

4000